Towards the Equation of State for Neutral (C2H4), Polar (H2O), and Ionic ([bmim][BF4], [bmim][PF6], [pmmim][Tf2N]) Liquids

Despite considerable effort of experimentalists no reliable vapor-liquid coexistence at very small pressures and liquid-solid coexistence at high pressures have been until now observed in the working range of temperature 290 < T/K < 350 for ionic liquids. The measurements of high-pressure properties in low-temperature stable liquid are relatively scarce while the strong influence of their consistency on the phase equilibrium prediction is obvious. In this work we discuss the applicability of fluctuationalthermodynamic methodology and respective equation of state to correlate the properties of any (neutral, polar, ionic) liquids since our ultimate goal is the simple reference predictive model to describe vapor-liquid, liquid-liquid, and liquid-solid equilibria of mixtures containing above components. It is shown that the inconsistencies among existing volumetric measurements and the strong dependence of the mechanical and, especially, caloric derived properties on the shape of the functions chosen to fit the experimental data can be resolved in the framework of fluctuational-thermodynamic equation of state. To illustrate its results the comparison with the known experimental data for [bmim][BF 4 ] and [bmim][PF 6 ] as well as with the lattice-fluid equation of state and the methodology of thermodynamic integration is represented. It corroborates the thermodynamic consistency of predictions and excellent correlation of derived properties over the wide range of pressures 0 < P/MPa < 200.


Introduction
Behavior of low-melting organic salts or ionic liquids (ILs) [1][2][3][4][5][6] in the region of phase transitions is qualitatively similar to that either for high-temperature nonorganic molten salts or long-hydrocarbon-chain organic solvents and, even, for polymer systems.Such characteristic features as negligible vapor pressure   (), undefined critical parameters   ,   ,   for vapor-liquid (V, )-transition, split of liquid-solid (l,s)boundary onto melting   () and freezing   () branches, existence of glassy states make the problem of metastability to be quite complex but vital for many potential uses of ILs.In particular, thermodynamic modeling and computer simulation of the phase behavior in mixtures formed by ILs with water and low-molecular organic solvents such as ethylene can be of great importance for the further tuning of their operational parameters.If one proceeds from a pure to a mixed fluid, it is especially advantageous to develop the same format of reference equation of state (EOS) and the common format of reference pair potential (RPP) for each component and mixture.
As a first step toward consistent modeling of the phase behavior of IL and its solution we demonstrate in this work how the fluctuational-thermodynamic (FT) EOS [7][8][9][10][11][12] and the relevant finite-range Len-nard-Jones (LJ) RPP can be applied to model the underlying structure and properties of low-molecular (C 2 H 4 , H 2 O) and imidazolium-based (1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF 4 ]), 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF 6 ]), 2,3-dimethyl-1-propylimidazolium bis(trifluoromethylsulfonyl)imide ([pmmim][Tf 2 N])) solvents.For any pure component FTmodel is based either on the measurable coexistence-curve input data   (),  V (),   () (if they are achievable as for C 2 H 4 and H 2 O) or on the also measurable one-phase density of liquid at atmospheric pressure  ( 0 ≈ 0,1 MPa, T) for ILs.This methodology becomes purely predictive for density (, ) in any one-phase V, , -regions including 2 Journal of Thermodynamics their metastable extensions.Only the measurable isobaric heat capacity data   ( 0 , ) have to be added to the set of input data for prediction of other caloric properties (isochoric heat capacity  V (, ), speed of sound (, ), and Gruneisen parameter Gr(, )) at higher pressures  >  0 and lower  <   or higher  >   temperatures where   is the hypothesized normal boiling temperature   ( 0 ).Its existence itself is a debatable question because the thermal decomposition   may be former   <   .
Such approach was proposed recently [7,8] to reconstruct the hypothetical (V, )-diagram of any ILs in its stable and metastable regions on the base of only standard reference data on density () at  0 [1][2][3][4] and one free parameter, an a priori unknown value of the excluded volume  0 .To our knowledge this is first attempt to predict simultaneously the whole set of one-phase and two-phase properties for ILs without the fit at any other pressures including the negative ones.It was argued that the particular low-temperature variant of the most general FT-EOS [9][10][11][12] should be used to obtain the consistent prediction of volumetric properties and the standard response functions   ,   ,   =   /  by the following equations: = (1 −  0 ) ⌊ −  2 (1 −  0 ) /⌋ 2 (1 −  0 ) 2 +  (2 0  − 1) , where  0 is the excluded molecular volume and () is the -dependent effective cohesive energy.The derivative / affects the thermal expansion   and the thermalpressure coefficient   while the isothermal compressibility   depends only on  0 -value at the given pressure.The changeable sign of two thermal derivatives   ,   offers a possibility to predict the properties of anomalous lowtemperature substances (such as water, for example) too [7,8].
Fortunately we have obtained now [13][14][15][16][17][18][19] a possibility to test our predictions not only by the direct experimental one-phase data [14,16,18,19] on (, )and (, )surfaces.Another possibility is offered by comparison of the predictions obtained by FT-EOS for the critical parameters of ILs ([bmim][BF 4 ]:   = 962,3 K,   = 3503,9 kPa,   = 438,565 kg⋅m −3 with those predicted here by the Sanchez-Lacombe EOS for lattice fluid (LF) [15]:   = 885,01 K,   = 2829 kPa,   = 248,565 kg⋅m −3 as well as with those simulated by GEMC-methodology [6]:   = 1252 K,   = 390 kPa,   = 181 kg⋅m −3 .It seems that the relatively close location of (  ,   )-parameters predicted by both EOSs is some guarantee of their reliability while   and   from [6] are significantly overestimated and underestimated, respectively.Interestingly, the known descriptive factor of compressibility   =   /(    ) estimated by Guggenheim [20] in the vicinity of triple point   for argon as   = 0,108 is equal to close values   = 0,082 for FT-EOS and   = 0,072 for LF-EOS but only to very small value   = 0,007 for result of GEMC-simulations if the common realistic estimate (see below)   ≈   = 5,350646 mol⋅dm −3 at T = 290 K is used.Moreover, it will be shown that the characteristic dimensional parameters  *  ,  ] provide the structural estimates of hard-core volume, number of lattice sites in a cluster, and energy of near-neighbor pair interactions which are surprisingly close to ones independently predicted by the FT-model of a continuum substance.
Taking into account the compatibility of above results it is important to consider the presumable similarity between the square-well fluid (which may be thought of as a continuum analogue of the lattice-gas (LG) or lattice-fluid (LF) systems) on the one hand and the LJ-fluid of finite-range interactions (RPP) on the other.This conceptual analogy has been pointed out long ago for the critical region by Widom [21] who suggested that it is the propagation of attractive correlations in the LG which determines the peculiarities of criticality.However, such unphysical LG-predictions at low temperatures of the (, )-plane as the nonexistence of a (, )transition suggest that repulsive forces are not being treated properly by this RPP-model.In contrast with the discrete LG-model, it seems that both attractive and repulsive forces are being dealt with properly in the square-well continuum fluid because it exhibits both (V, )and (, )-transitions.The serious restriction of latter is however evident since any singularities of RPP imply an artificial jump of pairdistribution isotropic function () at the point of cutoff radius   for attractive interactions.
In this context only the shifted and smoothed at  point LJ-potential [5,6] seems to be appropriate as RPP for a continuum system.Of course, the algebraic form of the respective reference EOS is essential too.In accordance with the statistical-mechanical arguments presented by Widom [21] there are the set of alternative forms including the original vdW-EOS and the LG-EOS in the well-known Bragg-Williams approximation which share the common restrictive feature.One may suppose that the probability of finding some prescribed value of the potential energy ( ⃗ ) at an arbitrary point in the fluid is independent of  at fixed : ( ⃗ ; ).Another simplifying assumption is that such EOS supposes only two types of fluid structure, one of the excluded (or hardcore) volume V 0 where the singular hard-sphere branch of potential is infinite and one of free volume ( − V 0 ) where the potential is uniform, weak, and unrestricted (an infiniterange rectilinear well).It should be directly proportional to density  = / = − where  is the total configurational energy and  is the constant vdW-coefficient.These historical notes are important to explain how one can go beyond the above restriction of -independency by adoption of linear dependence for a generalized specific or molar energy (see also (8) Another aim of the developed FT-EOS follows from the possibility [7] to estimate the effective LJ-parameters without any fit.Indeed, their general T-dependent values, are determined simply in the low-temperature range of all ILs where  0 is constant in (( 1)-( 4)) while the compressibility factor of saturated liquid   =   /(  ) becomes negligible as well as the vapor pressure   () trends to zero.Taking into account this asymptotic behavior it is especially important to study the possible correlations of these parameters in the RPP-model of an effective LJ-potential for ILs as the functions of total molecular weight .This concept is unusual for the conventional consideration of a separate influence of the anion's   and cation's   components.It may provide, in principle, the useful insight the nature of (V, )-transition in ILs by effective capturing underlying pair interactions.The distinction of both FT-EOS and LF-EOS [14] from the conventional hard-sphere reference EOS is crucial to provide the quantitative description of one-phase liquid.The formers include the quadratic in density contribution, which is dominating at high pressures along the isotherms.The latter considers this term as a small vdW-perturbation for the hard-sphere EOS.Such perturbation approach is not directly applicable to associating fluids such as water and alcohols for which presence of hydrogen bonding, anisotropic dipolar 1/ 3 , or coulombic 1/ interactions in addition to isotropic dispersive 1/ 6 attractions is inconsistent with the main assumption of the perturbation methodology that the structure of a liquid is dominated by repulsive forces [15].
The FT-model promotes the more flexible approach in which the above factors of attraction and clustering can be effectively accounted by the ()-dependence.It was firstly confirmed by Longuet-Higgins and Widom and, then, by many authors that a combination of Carnahan-Starling EOS, for example, with the vdW-perturbation  2 is a reasonable approximation for the and -phases but not the V-phase.Guggenheim [20] has concluded its applicability only to a liquid when large clusters are more important than small clusters (i.e., at low temperatures   <  <   ).In contrast with this observation, the general FT-EOS provides the adequate representation of entire subcritical range   <  <   including the critical region and (V, )-phase transition [9][10][11][12].It will be shown below by FT-model without undue complexity of calculations.

Universal FT-EOS for Any
Low-Temperature Fluids is applicable to any types of fluids including ILs.The measurable volumetric data of coexistence curve (CXC) have been used for evaluation of -dependences without any fit.Consider where the reduced slope   () of   ()-function is defined by the thermodynamic Clapeyron's equation: This fundamental ratio of the (V, )-latent heat to the thermodynamic work of (V, )-expansion is the main parameter of FT-coefficients determined by ((8)-( 9)).It should be calculated separately in each of high-temperature (  ≤  ≤   ) [9][10][11][12] v-and l-phases to obtain the reasonable quantitative prediction of one-phase thermophysical properties.The general FT-EOS is applicable to the entire subcritical range (  ≤  ≤   ) but it can be essentially simplified to the form of (1) if (  ≤  ≤   ).

Particular Form of FT-EOS for Low
Temperatures.An absence of input CXC-data   (),   (),   () for ILs is the serious reason to develop the alternative method for the evaluation of T-dependent FT-coefficients.The thermodynamically-consistent approach has been proposed in [7,8] for the particular form of FT-EOS (1)  To illuminate the distinction between the particular (reference) and general form of FT-EOS let us discuss in brief the main steps of the proposed procedure.Its detailed analysis can be found elsewhere [7,8].The algorithm is as follows.
Step 5. A preliminary value of  0 () may be estimated then by the more restrictive assumption  0 ≈ 0 (used also in the famous Flory-Orwoll-Vrij EOS developed for heavy nalkanes).Consider Step 6 (-variant of ()-prediction).To control the consistency of methodology one may use instead of Step 3 (variant) the same equation ( 17) with the approximate equality   () ≈  + () to solve (16b) at the a priori chosen  0 -value for determination of alternative (), and so forth, (Steps 4 and

5). Just this approach (B-variant) has been used below in the low-temperature range of [bmim][BF 4 ].
Step 7. The self-consistent prediction of a hypothetical (V, )-diagram requires the equilibration of CXC-pressures   (,   ) =   (,   ) by FT-EOS (1) with the necessary final change in  0 ()-value from (19) to satisfy the equality Only in the low-temperature range  ≤   the distinction between the preliminary definition (18) and its final form (19) for ()-values is not essential at the prediction of vapor pressure   (,   ).

Reference Equation of State, Effective Pair Potential, and Hypothetical Phase Diagram
To demonstrate universality of approach and for convenience of reader we have collected the coefficients of FT-EOS (1) for neutral (C 2 H 4 ) and polar (H 2 O) fluids [7,8] in Table 1 and  3).When temperature is low   <  <   FT-model follows a two-parameter ((),  0 ) correlation of principle of corresponding states (PCS) on molecular level as well as a two-parameter ((),  0 ) correlation of PCS on macroscopic level.
One the most impressed results of FT-methodology is shown in Figure 1 where the comparison between such different high-and low-molecular substances as ILs and C 2 H 4 , H 2 O is represented.The results based on the coefficients of Tables 1 and 2 demonstrate that the proposed lowtemperature model provides the symmetric two-value representation of vapor pressure ±  () similar to that observed for the ferromagnetic transition in weak external fields.
To estimate the appropriate excluded molar volume  0 (M = 225,82 g/mol) of FT-model we consider that it belongs to the range [V 0 = / 0 ≈ 162, V  = /  ≈ 187 cm 3 /mol].The extrapolated to zero temperature T = 0 K "cold" volume V 0 = 162 cm 3 /mol follows from (27).The fixed value:  0 = 178 cm 3 /mol ( 0 ≈ 1,1V 0 ) has been used in this work to demonstrate the main results of the proposed methodology.Such choice for [bmim][BF 4 ] on the ad hoc basis is in a good correspondence with the respective values:  0 = 195,3 cm 3 /mol for [bmim][PF 6 ] and  0 = 271,1 cm 3 /mol for [pmmim][Tf 2 N] where the empirical relationship  0 ≈ 1,1V 0 was also observed [7,8] with the tabular   ()-data for ethylene [23] and water [24]; (b) predicted two-value vapor-pressures rule for chain molecules (21b) usually considered by van der Waals'-type of EOS for mixtures [22].Consider The predicted by former rule of LJ-diameter for the same [bmim]-cation were close but still different, 5,906 Å and 5,682 Å.For the latter rule their values and distinction become even smaller, 5,757 Å and 5,651 Å.As a result, the chain rule (21b) seems preferable for ILs and its average value for    The collected in Table 4 effective LJ-diameters are linear functions of   in the set of ILs with different anions and cations if the molecular weight of latters   is the same one.
Since the low-temperature compressibility factor   () is very small for all discussed liquids their dispersive energies () (molecular attraction's parameters) are comparable in accordance with (6b).However, the differences in cohesive energies () (collective attraction's parameters) between the low-molecular substances (C 2 H 4 , H 2 O) and ILs are striking as it follows from Tables 1 and 2. The physical nature of such distinction can be, at the first glance, attributed to omitted in the reference LJ-potential influence of intramolecular force-field parameters and anisotropic (dipole-dipole and coulombic) interactions.At the same time, one must account the collective macroscopic nature of ()-parameter.It corresponds to the scales which are compatible or larger than the thermodynamic correlation length (, ).FT-model [9][10][11][12] provides an elegant and simple estimation of this effective parameter based on the concept of comparability between energetic and geometric characteristic of force field determined by the given RPP.Consider Taking into account the above results and the coefficients from Tables 1-3 we have used (22) at T = 300 K ( * =   / ≈ 1) to compare the thermodynamic correlation length predicted for [bmim][BF 4 ] (a = 8900,9 J⋅dm 3 /mol 2 ;  0 = 178 cm 3 /mol;  = 5,322294 mol/dm 3 ) and at T = 298,15 K for water (a = 548,27 J⋅dm 3 /mol 2 ;  0 = 16,58 cm 3 /mol; One may note that our estimates of correlation length are significantly larger than those usually adopted for the dimensional or reduced cutoff radius (  or  *  =   /) of direct interactions at computer simulations.As a result, the standard assumption  *  ≈  *  may become questionable in the comparatively small (mesoscopic) volumes of simulation  3 < (, ) 3 .At this condition the simulated properties are mesoscopic although their lifetime may be essentially larger than its simulated counterpart.The key point here is the same as one near a critical point where the problem of ].The characteristic (  ,   ) points are emphasized; tabular data for water [24] (-Q-).The location of spinodal predicted by LF-EOS [14] is shown by dashed line.consistency between the correlation length for statics and the correlation time for dynamics becomes crucial.In any case, the computer study of possible nongaussian nature of local fluctuations within the thermodynamic correlation volume  3 may be quite useful.The relevant inhomogeneities in the steady spacial distributions of density and enthalpy can affect, first of all, the simulated values of volumetric (  ,   ) and caloric (  ,  V ) derived quantities.Simultaneously, an account of internal degrees of freedom and anisotropy by the perturbed RPP may change the correlation length itself.
The above described by (( 12)-( 20)) FT-methodology has been used to reconstruct the hypothetical phase diagram (HPD) for [bmim][BF 4 ] shown in Figures 2, 3, and 4 and represented in Table 3.Both (, ) (Figure 2) and (, ) (Figure 4) projections contain also the branches of classical spinodal calculated by the LF (Sanchez-Lacombe)-EOS obtained in [14].Its top is the location of a respective critical point.It seems that the relatively close (  ,   )-parameters predicted independently by FT-EOS and by LF-EOS (see Section 1) are reasonable.
The FT-model provides a possibility to estimate, separately, the coordination numbers of LJ-particles in the orthobaric liquid   ()and vapor   ()-phases.An ability to form the respective "friable" ( , + 1)-clusters is defined by the ratio of effective cohesive and dispersive molar energies at any subcritical temperature.Consider The term "friable" is used here to distinguish the clusters formed by the unbounded LJ-particles at the characteristic distance  * = / ≈ 3 √ 2 > 1 from the conventional "compact" ones with the bonding distance  * < 1 studied, in particular, by the GEMC-methodology [25] to model of molecular association.It is straightforwardly to obtain the low-temperature estimates based on the assumptions.
and to find the critical asymptotics based on the difference of classical ( 0 ,  0 ,  0 ) and nonclassical (, , ) -dependent FT-EOS' coefficients [9][10][11][12].Consider The crucial influence of excluded-volume in (24a) and its relative irrelevance in (24b) for  , -predictions are illustrated by Figure 5 where   () function is shown also for the entire l-branch based on the evaluated in the present work HPD.For comparison, the low-temperature ability to form the (  + 1)-clusters in liquid water [7,24] is represented in Figure 5 too.
In according with ((25a), (25b)) the "friable" clusters can exist only as dimers in the classical critical liquid phase (   ≈ 1).It is not universal property in the meaning of scaling theory but it corresponds to the PCS-concept of similarity between two substances (H 2 O and [C 4 mim][BF 4 ], e.g.) if their   -values are close.On the other side, the scaling hypothesis of universality is confirmed by the FT-EOS' estimates in the nonclassical critical vapor phase.For the set of low-molecular-weight substances studied in [9] (Ar, C 2 H 4 , CO 2 , H 2 O); for example, one obtains by (25b) the common estimate (   ≈ 2,5) which shows a significant associative near-mean-field behavior.
It is worthwhile to note here the correspondence of some FT-EOS'-estimates with the set of GEMC-simulated results.One may use the approximate estimate of critical slope   ≈ 7,86 [9] for [bmim][BF 4 ] based on the similarity of its   -value with that for H 2 O [24].In such case, the respective critical excluded volume   ≈ 220 cm 3 /mol becomes much more than vdW-value 1/3  =  0  ≈  0 ≈ 178 cm 3 /mol.Another observation seems also interesting.Authors [25] have calculated (see Figure 3 in [25]) for the "compact" clusters at  * = 0,7; 0,5; 0,45; the ( * ,  * )-diagram of simple fluids.One may note that only the value  * = 0,7 corresponds to the shape of strongly-curved diameter shown in Figure 2 for the HPD while the smaller values:  * = 0,5; 0,45 give the shape of HPD and the nearly rectilinear diameter strongly resembling those obtained by the GEMCsimulations [6] for the complex IL's force-field.If this correspondence between the "friable" and "compact" clustering is not accidental one obtains the unique possibility to connect the measurable thermophysical properties with the both characteristics of molecular structure in the framework of FT-EOS.

Comparison with the Empirical Tait EOS and Semiempirical Sanchez-Lacombe EOS
The empirical Tait EOS is based on the observation that the reciprocal of isothermal compressibility  −1  for many liquids is nearly linear in pressure at very high pressures.Consider where some authors [14,19]  Two other reasons of discrepancies in the Tait methodology is the different approximations chosen by authors for the reference input data ( 0 , ) and for the compounddependent function ().Some authors [4,14,18] prefer to fit the atmospheric isobars ( 0 , ) and   ( 0 , ) with a second-order or even third-order polynomial equation while the others [1,2,16,19] use a linear function for this aim..As a result, the extrapolation ability to lower and higher temperatures of different approximations becomes restricted.
The different choices of an approximation function for () (so authors [14] have used the exponential form while authors [19] have preferred the linear form) may distort the derivatives   and   calculated by the Tait EOS (26).The problem of their uncertainties becomes even more complex if one takes into account the often existence of systematic distinctions of as much as 0,5% between the densities measured by different investigators even for the simplest argon [23].Machida et al. [14], for example, pointed out the systematic deviations measured densities from those reported by the de Azevedo et al. [18] and Fredlake et al. [1] for both [bmim][BF 4 ] and [bmim][PF 6 ].Matkowska and Hofman [19] concluded that the discrepancies between the different sets of calculated   -and   -derivatives increase with increasing of  and decreasing of  due not only to experimental differences in density values but also result from the fitting equation used.The resultant situation is that the expansivity   of ILs reported in literature was either nearly independent of T [18] or noticeably dependent of T [3,19].
We can add to these observations that the linear in molar (or specific) volume Tait Eos (26) is inadequate in representing the curvature of the isotherm () at low pressures.It fails completely in description of (V, )-transition where the more flexible function of volume is desirable.However, this has been clearly stated and explained by Streett for liquid argon [23] that the adjustable T-dependence of empirical EOS becomes the crucial factor in representing the expansivity   and, especially, heat capacities   ,  V at high pressures even if the reliable input data of sound velocity (, ) were used.
From such a viewpoint, one may suppose that the linear in temperature LF-EOS proposed by Sanchez and Lacombe, is restricted to achieve the above goal but can be used as any unified classical EOS common for both phases to predict the region of their coexistence.Such conjecture is confirmed by the comparison of FT-EOS with LF-EOS presented in Figures 2 and 4 and discussed below.The obvious advantage of former is the more flexible T-dependence expressed via the cohesiveenergy coefficient ().On the other hand, the LF-EOS is typical form of EOS (see Section 1) in which the constraint of T-independent potential energy ( ⃗ ; ) is inherent [21].One may consider it as the generalized variant of the wellknown Bragg-Williams approximation for the ordinary LG presented here in the dimensional form Such generalization provides the accurate map of phenomenological characteristic parameters  * ,  * ,  * which determine the constant effective number of lattice sites   occupied by a complex molecule, into the following set of molecular characteristic parameters for a simple molecule (  = 1): where V 0 is the volume of cell and  is the coordination number of lattice in which the negative  is the energy of attraction for a near-neighbor pair of sites.In the polymer terminology  * from (31) is the segment interaction energy and V * is the segment volume which determines the characteristic hard core per mole / * (excluded volume  in the vdW-terminology).Another variant of described approach is the known perturbed hard-sphere-chain (PHSC) EOS proposed by Song et al. [15] for normal fluids and polymers where () = (1 − /2)/(1 − ) 3 is the pair radial distribution function of nonbonded hard spheres at contact and the term with (  − 1) reflects chain connectivity while the last term is the small perturbation contribution.Though the PHSC-EOS has the same constraint of the potential energy field ( ⃗ ; ) authors [15] have introduced two universal adjustable Φ  ()and Φ  ()-functions to improve the consistency with experiment.The vdW-type coefficients were rescaled as where (  ) is the additional scaling function for  * = /  .It provides the interconnection of molecular LJ-type parameters (, ) with the phenomenological vdW-ones (, ).The resultant reduced form of PHSC is [15] where the following characteristic and reduced variables are used: It was compared with the simpler form of LF-EOS (29).
Their predictions of the low-temperature density at saturation   () are comparable but, unfortunately, inaccurate (overestimated) even for neutral low-molecular liquids.The respective predictions of the vapor pressure   () are reasonable [15] excepting the region of critical point for both EOSs.Our estimates based on the LF-EOS [14] shown in Figures 2 and 4 are consistent with these conclusions.The comparison of volumetric measurements and derived properties [14,18] with the purely predictive (by the FT-EOS) and empirical (by the Tait EOS and LF-EOS) methodologies used for [bmim][BF 4 ] is shown in Figures 6-9.Evidently, that former methodology is quite promising.Machida et al. [14] have reported two correlations of the same (, , )data measured for [bmim][BF 4 ] at temperatures from 313 to 473 and pressures up to 200 MPa.To examine the trends in properties of ILs with the common cation [bmim] the Tait empirical EOS was preliminarily fitted as the more appropriate model.The estimate of its extrapolation capatibilities for (, , )-surface in the working range (290 < /K < 350) follows from the compatibility of experimental points (where those measured by de Azevedo et al. [18] in the range of temperature 298 < /K < 333 and pressure (0,1 < /MPa < 60) were also included) with the thick curves in Figure 6.It is noticeable, for example, that the extrapolated Tait's isotherm T = 290 K coincides practically with isotherm T = 298,34 K from [18] because the measured densities of latter source are systematically higher than those from [14].Density data of Fredlake et al. [1] for [bmim][BF 4 ] (not shown in Figure 6) are also systematically shifted from measurements [14].(l-298,34 [18]; ◻-313,01 [18]; △-322,85 [18]; ♢-332,73 [6]; ◼-313,1 [14]; Q-332,6 [14]; -352,6 [14]) with those calculated: (a) by the Tait EOS [14] (in a working range 290 < /K < 350 via the interval 10 K thick continuous curves); (b) by the Sanchez-Lacombe EOS [14] (thick dashed curves); (c) by the FT-EOS (thin continuous curves).The consequence of such discrepancies is also typical for any simple liquids (Ar, Kr, Xe) [23] at moderate and high pressures.It is impossible to reveal an actual T-dependence of volumetric (mechanical) derived functions   ,   due to systematic deviations between the data of different investigators.In such situation an attempt "to take the bull by the horns" and to claim the preferable variant of EOS based exclusively on volumetric data may be erroneous.Indeed, since the Tait EOS is explicit in density while the LF-EOS-in temperature the direct calculation of   ,   -derivatives for former and   ,   -derivatives for latter are motivated.To illustrate the results of these alternative calculations we have used in Figures 6-9 the coefficients of LF-EOS reported by Machida et al. [14] for the restricted range of moderate pressures 0,1 < /MPa < 50.The thick dashed curves represent the boundaries of working range where the extrapolation to T = 290 K is again assumed.One may notice the qualitative similarity of FT-EOS (the thin curves) and LF-EOS which can be hypothesized as an existence of certain model substance at the extrapolation to higher pressures /MPa > 50.It demonstrates the smaller compressibility   (Figure 8) and expansivity   (Figure 7) than those predicted by the Tait EOS while the value of thermal-pressure coefficient   for FT-EOS (Figure 9) becomes larger.It determines the distinctions in the calculated internal pressure.The choice of the FTmodel's substance as a reference system for the perturbation methodology provides the set of advantages in comparison with the LF-EOS.It follows from Figure 6 that at moderate pressures /MPa < 50 the predictive FT-EOS is more accurate than the fitted semiempirical LF-EOS [14] although the discrepancies of both with the empirical Tait EOS [14] become significant at the lowest (extrapolated) temperature T = 290 K.The Tait's liquid has no trend to (V, )-transition (as well as polymers) in opposite to the clear trends demonstrated by FT-EOS and LF-EOS.One may suppose [26] a competition between vaporization of IL (primarily driven by the isotropic dispersive attraction 1/ 6 in RPP) and chain formation (driven mainly by the anisotropic dipolar interactions 1/ 3 ) reflected by the Tait EOS fitted to the experimental data.Of course, such conjecture must be, at least, confirmed by the computer simulations and FT-model provides this possibility by the consistent estimate of RPP-parameters (, ) at each temperature.
The differences of calculated expansivity   in Figure 7 are especially interesting.FT-EOS predicts even less variation of it with temperature than that for the Tait EOS.This result and crossing of   ()-isotherms are qualitatively similar to those obtained by de Azevedo et al. [18] although the pressure dependence of all mechanical (  ,   ,   ) and caloric (  ,  V ) derivatives (see Figures 10,11,and 12) is always more significant for the FT-EOS predictions.It seems that the curvature of the ()-dependence following from the LF-EOS (29) is not sufficient to predict the   () behavior (Figure 7) correctly.The strong influence of the chosen input ( 0 , )-dependence is obvious from Figures 7-9.
The prediction of caloric derivatives (  ,  V ,   / V , Gr) is the most stringent test for any thermal (, , ) EOS.It should be usually controlled [23] by the experimental (, , )-surface to use the thermodynamical identities,  13) with the evaluated at high pressures heat capacities.Our predictive strategy is based [17] on the differentiation of ()-dependence to evaluate directly the most subtle ( V , , )-surface in a low-temperature liquid where the influence of the consistence for the chosen input ( 0 , )and   ( 0 , )-dependences (via (37) used for estimate of  V ( 0 , ) at the atmospheric pressure  0 ) becomes crucial.The use of first derivative / (even by its rough approximation in terms of finite differences: Δ/Δ) to ] (thin lines) with that (points) used by de Azevedo et al. [18] as the input data (see also Figure 5 for density used by de Azevedo et al. [18] as the input data at evaluations).
calculate simultaneously by ((3), ( 4), (37), ( 40)) all volumetric and caloric derivatives is the important advantage over the standard integration of thermodynamic identities: To illustrate such statement it is worthwhile to remind the situation described by the Streett [23] for liquid argon.Since isotherms of   () cross over for many simple liquids (Ar, Kr, Xe), this author concludes that the sign of ( 2 V/ 2 )  changes also from positive to negative at the respective pressure.This conclusion is not valid because to change the sign of derivative (  /)  it is enough to account for the exact equality in which ( 2 V/ 2 )  can be always positive.In this case one would expect the monotonous decrease of   with increasing  in accordance to ((41a), (41b)) while the presence of extremum (minimum or maximum of   ()-dependence) seems to be artificial.There is the variety of pressures reported by different investigators as a presumable cross-point for the same ILs.Machida et al. [14] have estimated it to be about 10 MPa on the base of Tait EOS for [bmim][PF 6 ] but have not found it (Figure 7) for [bmim][BF 4 ].For latter our estimate by the FT-EOS is: P = 20,6 MPa. de Azevedo et al. [18] have reported the mild decrease of   ()-dependence and the sharp decrease   ()-dependence while a presumable cross-point is located between about 100 and 120 MPa for [bmim][BF 4 ].Taking into account the above distinction in the evaluated (  , , )surface it is interesting to consider their consequences for  The remarkable qualitative and even quantitative (<8%) correspondence between the predicted by FT-EOS  V -values and those reported by de Azevedo et al. [18] follows from Figure 10.At the same time, although the discrepancies between   -values [18] and those predicted by the FT-EOS are again within acceptable limits (<10%) the formers demonstrate the weak maximum and very small pressure dependence for [bmim][BF 4 ] (for [bmim][PF 6 ] this   ()dependence is monotonous as well as that predicted by the FT-EOS).It seems that the resultant ratio of heat capacity   / V shown in Figure 12 which demonstrates the irregular crossing of isotherms [18] is questionable.It suggests that their pressure dependence either needs the more accurate approximation or reflects the realistic distinction of reference FT-EOS from the actual behavior of [bmim][BF 4 ].
The lock of noticeable variations in pressure is the common feature of integration methodology [18] based on the given (, , )and (, , )-surfaces.The unavoidable accumulation of uncertainties at each stage of calculations in the set  −   −  V may cause the unplausible behavior of adiabatic exponent   / V in liquid.The same is true for the set  V −   −  used in the FT-methodology.It is the most appropriate explanation of significant discrepancies for ()-dependence shown in Figure 13.Let us remind also that the precise mechanical measurements of speed velocity [18] in the very viscous IL cannot be attributed exactly to the condition of constant entropy.Thus, strictly speaking, the measured (, , )-surface reflects the strong dispersive properties of media and must be less than its thermodynamic counterpart in the ideal (without a viscosity) liquid.

Conclusions
There are the structure-forming factors related to the abovediscussed thermodynamic characteristic.Despite the certain discrepancies between the predicted and derived properties for FT-EOS and LF-EOS, both ones provide the close estimates of structure factors represented in Table 5.
Our aim here is to show that the thermodynamicallyconsistent predictions of thermodynamical properties by the FT-EOS yields also the molecular-based parameters which are, at least, realistic (see also Table 3).The estimate of average T-dependent well-depth  by (6b) as well as estimate of average value   by (24a) is related to the middle of temperature range: T = 320 K.The distinction of  from the respective  * -parameters of LF-EOS [14] can be attributed to the difference between nonbonded interactions in the discrete (LF-EOS) and continuum (FT-EOS) models of fluid.Our estimate of cohesive-energy density  coh by equality,  coh =  2 ( 0 , )  () +   ()   () , represented in Table 6 seems also physically plausible.Maginn et al. [5,6] have determined it within the framework of GEMC-simulations by the knowledge of ( 0 , ) and

Figure 5 :
Figure 5: Comparison of two coordination numbers   predicted by FT-EOS to characterize the clustering in an orthobaric liquid of [bmim][BF 4 ] at  ≤   (◼zz) and in the entire (V, )-range of HPD (-◼-◼); FT-EOS predictions for water are shown as -Q at  ≤ 373 K.

Figure 13 :
Figure 13: Comparison of predicted by FT-EOS speed of sound for [bmim][BF 4] (thin lines) with that (points) used by de Azevedo et al.[18] as the input data (see also Figure5for density used by de Azevedo et al.[18] as the input data at evaluations). .

Table 6 :
[16]arison of internal pressure (/V)  for [bmim][BF 4 ] based on the LF-EOS[14]and FT-EOS (this work) with the values estimated[16]by experimental data on speed of sound , density , and isobaric heat capacity   .